NONSTANDARD METHODS IN RAMSEY’S THEOREM
FOR PAIRS
C. T. CHONG
Abstract. We discuss the use of nonstandard methods in the
study of Ramsey type problems, and illustrate this with an example
concerning the existence of definable solutions in models of BΣ02
for the combinatorial principles of Ramsey’s Theorem for pairs and
cohesiveness.
1. Introduction
Ramsey’s Theorem (RTnk ) states that every partition of the n-element
subsets of N into k classes has an infinite homogeneous set. The prooftheoretic strength of RTnk is an area of active research by recursion
theorists in recent years.
Taking the system of second order arithmetic RCA0 as the base
theory, an earlier result of Jockusch [9] implies that for n > 2, the
arithmetic comprehension axiom is a consequence of RTnk . This was
shown not to hold under RT22 by Seetapun and Slaman [12], so that
over RCA0 , RT22 is strictly weaker than RTnk for n > 2.
In a systematic study of RT22 , Cholak, Jockusch and Slaman [1] introduced two combinatorial principles related to RT22 , defined over a
model M = hM, X, +, ·, 0, 1i of RCA0 , where X is the collection of
second order objects of M:
(1) The principle of stable Ramsey’s Theorem for pairs (SRT22 ): A
partition f of [M ]2 into two classes (sometimes also called “two
colors”) is said to be stable if lims f (x, s) exists for all x ∈ M .
SRT22 states that every stable partition of [M ]2 into two classes
has an infinite homogeneous set in M.
1991 Mathematics Subject Classification. 03D20, 03F30, 03H15.
The author wishes to thank the Department of Mathematics, University of California, Berkeley, for its hospitality during his visit from October to December 2005,
where part of the research was done. This paper is a revised version of a talk given
in August 2005 at the Program in Computational Prospects of Infinity held at the
Institute for Mathematical Sciences, National University of Singapore .
1
2
C. T. CHONG
(2) The principle of cohesiveness (COH): Let Y be an M -infinite
set in M and let Ys = {t|(s, t) ∈ Y }. hYs is∈M is called a Y array. Then a set C is Y -cohesive if for all s, either C ∩ Ys is
M-finite or C ∩ Y¯s is M-finite. COH states that every Y -array,
where Y ∈ M, has a Y -cohesive set C ∈ M.
It is known ([1]) that over the base theory RCA0 , Ramsey’s Theorem for pairs is equivalent to SRT22 + COH. In view of the definition
of stable 2-coloring, it is natural to conjecture that SRT22 is strictly
weaker than RT22 . Much effort has been expended on establishing this
conjecture. First of all, it is not difficult to see that, given a stable
partition f ∈ M of [M ]2 into two colors, there is an f -homogeneous
set ∆02 in the parameter defining f . Jockusch [9] exhibited a recursive
partition fJ of [N]2 into two classes that has no ∆2 homogeneous solution (thus the partition is necessarily nonstable). A first attempt to
proving the relative strength of SRT22 and RT22 is to show that every
stable recursive partition of [N]2 into two colors has a low homogeneous solution (i.e. whose jump is ∅0 ). An iteration of the construction
will then produce a model of RCA0 + SRT22 that does not include a
homogeneous solution to fJ according to Jockusch’s result. Downey,
Hirschfeldt, Lempp and Solomon [6] showed that this approach failed
by demonstrating the existence of a ∆2 set A of integers with no infinite
low subset in A or Ā. Thus to produce a model M = hM, X, +, ·, 0, 1i
of RCA0 + SRT22 without RT22 where M = N (i.e. an ω-model) requires
a more sophisticated approach. In [7], Hirschfeldt, Jockusch, KjosHanssen, Lempp and Slaman show that while a low homogeneous set
is not always guaranteed for a stable 2-coloring of pairs, it is nevertheless true that an incomplete ∆2 homogeneous solution exists. The
authors also propose ways of attacking the SRT22 problem by considering different possible ω-models that could be constructed to avoid
RT22 .
In this paper we discuss the use of nonstandard models in investigations of Ramsey type problems. From the proof-theoretic point of
view, there is no reason to restrict oneself to ω-models in studying
subsystems of second order arithmetic. Since there exist nonstandard
models of RCA0 that exhibit simpler structures in the Turing degrees,
Ramsey type problems may be analyzed from a different perspective
and this could shed light on the problems themselves. For example,
there exist models of RCA0 + BΣ02 in which every incomplete Turing
degree is low. This means that the counterexample given in [6] does
not apply to these models. It will be useful to study the problem of
SRT22 in such a situation. This is discussed further later in the paper.
NONSTANDARD METHODS IN RAMSEY’S THEOREM FOR PAIRS
3
Our main result gives an illustration of the link between Ramsey type
problems and nonstandard models. Namely, we are concerned with the
question of existence of solutions for RT22 or COH that is recursive in
the double Turing jump of the set parameter defining the 2-coloring or
array. Classical results of Jockusch and others show that this is true
for ω-models. In general, however, the answer is determined by the
underlying first order inductive strength of the model being considered
(Corollaries 3.1, 3.2 and 3.3).
We assume that the reader is familiar with recursion theory in models
of fragments of Peano arithmetic as well as subsystems of second order
arithmetic (cf. [4] and [14] for details). Here we will only present a
summary of the facts that are relevant to the discussion below.
2. BΣ02 models
Let IΣ0n denote the induction scheme for Σ0n formulas (with number
and set parameters), where n ≥ 0. All models M considered in this
paper satisfy at least IΣ01 (IΣ0n is denoted Σ0n -IND in [14]). A bounded
set in M is M-finite if it is coded in M. Otherwise it is called Minfinite. An unbounded set in M is necessarily M-infinite, although
the converse is not always true (for example N is not M-finite in any
nonstandard model M).
Let BΣ0n denote the scheme which states that every Σ0n definable
function maps an M-finite set onto an M-finite set. The result of
Kirby and Paris [10] for first order formulas may be generalized to
show that for all n ≥ 0, IΣ0n+1 is strictly stronger than BΣ0n+1 , which
is in turn strictly stronger than IΣ0n . Indeed, the results stated without
proof in this section were originally shown for first order fragments of
Peano arithmetic. Their generalization to subsystems of second order
arithmetic is immediate.
Proposition 2.1. If M |= IΣ0n , then every bounded Σ0n (M) set is
M-finite.
If M |= BΣ0n , then a cut I ⊂ M is a set that is closed downwards as
well as under the successor function. I is a Σ0n cut if it is Σ0n definable
over M. Proposition 2.1 implies that M |= IΣ0n if and only if there is
no proper (i.e. bounded) Σ0n cut. We consider only proper Σ0n cuts in
this paper.
If M |= BΣ0n but not IΣ0n , we call it a BΣ0n model. In this case,
there is a Σ0n (M) function mapping a Σ0n cut cofinally into M .
We identify a number in M with the set of its predecessors.
4
C. T. CHONG
Definition 2.1. Let M be a model of RCA0 . A set X ⊂ M is regular
if X ¹ a is M-finite for each a ∈ M .
Proposition 2.2. Let M be a BΣ02 model of RCA0 . Then
(i) Every ∆02 (M) set is regular;
(ii) X is ∆02 (M) if and only if X is recursive in Y 0 , where Y is the
set parameter occurring in the definition of X.
(iii) Every X ∈ X is regular.
Proposition 2.3. Let M be a model of RCA0 and X ∈ X. Then
M |= BΣ02 (X) if and only if M |= BΣ01 (X 0 ).
Definition 2.2. Let M |= RCA0 and X ⊂ M . Then X is hyperregular if the image of every bounded set under a function that is weakly
recursive in X (i.e. ∆1 (X)) is bounded.
Given a model M of RCA0 , and A ⊂ M , let M[A] denote the
structure generated from A over M by closing under functions recursive
in A ⊕ X, where X ∈ X.
Proposition 2.4. (Mytilinaios and Slaman [11]) Let M be a BΣ02
model of RCA0 and A ⊂ M . The following are equivalent:
(i) A is hyperregular;
(ii) M[A] |= IΣ01 .
The notion of coding in BΣ02 models of RCA0 is central to the analysis
of Ramsey type problems in this paper. Although the original definition was given for first order structures, it carries over to second order
structures in an obvious way.
Definition 2.3. Let A be a subset of M , where M |= RCA0 . A set
X ⊆ A is coded on A if there is an M-finite set X̂ such that X̂∩A = X.
Definition 2.4. Let A be a subset of M. We say that a set X is ∆0n
on A if both A ∩ X and A ∩ X are Σ0n (M).
The following result of Chong and Mourad [2] will be used repeatedly
in the sequel:
NONSTANDARD METHODS IN RAMSEY’S THEOREM FOR PAIRS
5
Proposition 2.5. If M is a BΣ0n model of RCA0 , then every bounded
set that is ∆0n (M) on a given set A is coded on A.
3. Ramsey’s Theorem in BΣ02 models of RCA0
We first prove a general theorem for BΣ02 models of RCA0 , and then
use it to study Ramsey’s Theorem for pairs and COH in such models.
From now on, let M = hM, X, +, ×, 0, 1i be a model of RCA0 + BΣ02
with a Σ02 cut I. Also fix g : I → M to be an increasing Σ02 cofinal
function, with set parameter Y ∈ X.
Theorem 3.1. If C ≤T Y 00 is in X, then C 0 ≤T Y 0 .
We first prove a lemma.
Lemma 3.1. If C ≤T Y 00 is in X, then (C ⊕ Y )0 ≤T I ⊕ Y 0 .
00
Proof. Let ΦY = C. We identify a set with its characteristic function.
By Proposition 2.2 (iii), C is regular. Hence for each i ∈ I, there is a
neighborhood condition hPi , Ni i of Y 00 such that hC ¹ g(i), Pi , Ni i ∈ Φ.
Claim 1. (C ⊕ Y )0 is regular.
Proof of Claim 1. Since C and Y are in X, BΣ02 (C ⊕ Y ) holds in M.
Fix i ∈ I. For x ∈ g(i) ∩ (C ⊕ Y )0 , let h(x) be the least y such that
x is in (C ⊕ Y )0y , the set enumerated into (C ⊕ Y )0 by stage y using
C ⊕ Y as an oracle (recall that (C ⊕ Y )0 is Σ1 (C ⊕ Y )), and let it be 0
otherwise. Then by BΣ02 (C ⊕ Y ), the image of g(i) under h is M-finite
and has a largest element denoted y0 . Then for x < g(i), x ∈ (C ⊕ Y )0
if and only if x ∈ (C ⊕ Y )0y0 .
Claim 2. If D ⊂ Y 00 is M-finite, then there is a jD such that D ⊂
00
00
Yg(j
, where Yg(j
is the set of elements enumerated into Y 00 by stage
D)
D)
g(jD ) using Y 0 as oracle.
Proof of Claim 2. By Proposition 2.3, BΣ01 (Y 0 ) holds. The map
taking each x ∈ D to the least y such that x ∈ Yy00 is Y 0 -recursive,
hence M-finite and bounded below a y0 . Let jD be the least j ∈ I such
that g(j) > y0 .
Claim 3. If X ≤T Y 00 is regular, then X ≤T I ⊕ Y 0 .
6
C. T. CHONG
00
Proof of Claim 3. Suppose ΨY = X. Then for each i ∈ I, there
exists hPi , Ni i such that Pi ⊂ Y 00 and Ni ⊂ Ȳ 00 with hX ¹ g(i), Pi , Ni i ∈
0
00
Ψ. By Claim 2, there is a j ∈ I such that Pi ⊂ Yg(j)
and Ni ⊂ Ȳg(j)
.
Hence the set
(1)
00
00
&
& N ⊂ Ȳg(j)
Z = {(i, j, j 0 ) ∈ I × I × I|∀hP, N i[P ⊂ Yg(j)
00
ΨhP,N i ¹ g(i) total → (N ∩ Yg(j
0 ) 6= ∅]}
satisfies the property that for each i, the set {(j, j 0 )|(i, j, j 0 ) ∈ Z} is
bounded in I × I. Furthermore, Z is ∆02 (Y ) on I × I × I and therefore
coded on I × I × I by an M-finite set Ẑ according to Proposition 2.5.
We may assume that for each (i, j), j 0 is the least such that (i, j, j 0 ) ∈ Ẑ.
¯
Let ji be the greatest j for (i, j, j 0 ) ∈ Z. Then for ji + 1, (ji + 1)0 ∈ I.
00
00
In particular, there is a hP, N i ⊂ Yg(ji +1) × Ȳg(ji +1) such that N ⊂ Ȳ 00
and ΨhP,N i ¹ g(i) is total, and so equal to X ¹ g(i). Furthermore, since
X ¹ g(i) is M-finite, all the computations in Ψg(ji +1) that disagree with
X ¹ g(i) will be identified at stage g(j ∗ ) for some j ∗ > ji . The least
hP,N i
such j ∗ , denoted ji∗ , is where there is only one output for Ψg(j ∗ ) ¹ g(i)
00
00
for any hP, N i ∈ Yg(j
, Ȳg(j
.
i +1)
i +1)
Now we may compute C from I ⊕ Y 0 as follows: To decide C ¹ y, use
Y 0 to choose an i such that g(i) > y. Use I to find the least j such that
¯ This is ji + 1. Then C ¹ g(i) is the set D such
(i, j, j 0 ) ∈ Ẑ and j 0 ∈ I.
00
00
that there is an N ⊂ Ȳg(j ∗ ) with hD, Yg(j
, N i ∈ Ψg(ji +1) ∩ Ψg(ji∗ ) .
i +1)
i
By Claim 1 and Claim 3, C ⊕ Y ≤T I ⊕ Y 0 . Since BΣ02 (C ⊕ Y )
holds, IΣ01 (C ⊕ Y ) is true and therefore by Proposition 2.4, C ⊕ Y is
hyperregular. This means that for all i, there is a least j, denoted ĵi ,
such that (C ⊕ Y )0 ¹ g(i) = (C ⊕ Y )0g(j) ¹ g(i). Now the set
V = {(i, j)|(C ⊕ Y )0g(j) ¹ g(i) 6= (C ⊕ Y )0g(j+1) ¹ g(i)}
is ∆02 (C ⊕ Y ) on I × I and therefore coded on I × I by an M-finite
set V̂ according to Proposition 2.5. Furthermore, ĵi is the least j ∈ I
such that (i, j) ∈
/ V . Then using V̂ as a parameter, we may compute
(C ⊕ Y )0 from I ⊕ Y 0 as follows: Given i, use I to obtain ĵi and then
use Y 0 to compute (C ⊕ Y )0g(ĵ )) . Then x ∈ (C ⊕ Y )0 ¹ g(i) if and only
i
¤
if x ∈ (C ⊕ Y )0g(ĵ ) . This completes the proof of the lemma.
i
Proof of Theorem 3.1
NONSTANDARD METHODS IN RAMSEY’S THEOREM FOR PAIRS
7
Proof. First note that just like Claim 1 of Lemma 3.1, one can show
0
that Y 0 is regular. By Lemma 3.1 let ΨI⊕Y = (C⊕Y )0 . A neighborhood
condition hP, N i of I may be identified with a pair (c, d) where c ∈ I
and d ∈ I¯ (let c be the maximum of P and d be the minimum of N ).
We make the following claim.
Claim. For each i ∈ I there exist (j, j 0 , j 00 ) ∈ I × I × I such that
0
00
(i) Existence. There is a (c, d) with c < j < j 0 ≤ d and Ψ(c,d)⊕Y ¹g(j ) ¹
0
g(i) = ΨI⊕Y ¹ g(i) and
(ii) Consistency. There is no x < g(i) and (c0 , d0 ) with c0 ≤ j <
0 0
0
00
0
00
j 0 ≤ d0 such that Ψ(c ,d )⊕Y ¹g(j ) (x) ↓6= ΨI⊕Y ¹g(j ) (x).
Proof of Claim. Otherwise, there is an i ∈ I such that for all
(j, j 0 , j 00 ) ∈ I × I × I, either (i) or (ii) is false. More precisely, there is
an i ∈ I such that for all (j, j 0 , j 00 ) ∈ I × I × I, either
0
00
(iii) For any (c, d) with c ≤ j < j 0 ≤ d, if Ψ(c,d)⊕Y ¹g(j ) ¹ g(i) ↓, then
0
00
0
00
there is an x < g(i) such that Ψ(c,d)⊕Y ¹g(j ) (x) 6= ΨI⊕Y ¹g(j ) (x),
or
(iv) There is an x < g(i) and a (c, d) with c ≤ j < j 0 ≤ d such that
0
00
0
00
Ψ(c,d)⊕Y ¹g(j ) (x) ↓6= ΨI⊕Y ¹g(j ) (x) (in this case d has to be in I).
Since (C ⊕ Y )0 ≤T I ⊕ Y 0 , there exist (j0 , j00 , j000 ) in I × I × I and
0
0
00
c0 ≤ j0 < j00 < d0 ∈ I¯ such that Ψ(c0 ,d0 )⊕Y ¹g(j0 ) ¹ g(i) = ΨI⊕Y ¹
g(i). Hence for the triple (j0 , j00 , j000 ), (iii) is false and (iv) must hold.
Now if the set J = {j|(iv) holds for (j0 , j, j000 ) with c ≤ j0 < j ≤ d} is
bounded in I, say by j ∗ , then it implies that for any (c, d) such that
0
00
0
00
c ≤ j0 < j ∗ ≤ d, if Ψ(c,d)⊕Y ¹g(j0 ) (x) ↓ then it is equal to ΨI⊕Y ¹g(j0 ) (x).
¯ we see that (j0 , j ∗ , j 00 ) satisfies (i) and (ii),
Since c0 < j ∗ < d0 ∈ I,
0
contradicting the assumption that i is a counterexample to the claim.
Hence J is unbounded in I. But then this implies that for d < d0 ,
d ∈ I¯ if and only if for all d0 where d ≤ d0 ≤ d0 , there is no c ≤ j0
0
0
00
0
00
and x < g(i) such that Ψ(c,d )⊕Y ¹g(j0 ) (x) ↓6= Ψ(c0 ,d0 )⊕Y ¹g(j0 ) (x). This
gives I¯ a Π01 (Y 0 ) definition (since Y 0 ¹ g(j000 ) and (C ⊕ Y )0 ¹ g(i) are both
M-finite), hence M-finite, which is a contradiction, proving the Claim.
By the Claim, the set of quadruples (i, j, j 0 , j 00 ) satisfying (i) and (ii)
is ∆02 (C ⊕Y ) on I ×I ×I ×I, hence coded on I ×I ×I ×I by an M-finite
set ĥ which we may consider to be a function taking i to the least triple
(j, j 0 , j 00 ). Then Y 0 computes (C ⊕Y )0 as follows: Given y, find an i ∈ I
using Y 0 such that g(i) > y. Now use hĥ1 (i), ĥ2 (i)i ⊕ Y 0 ¹ g(ĥ3 (i)) to
enumerate (C ⊕ Y )0 ¹ g(i) via Ψ, where ĥ(i) = hĥ1 (i), ĥ2 (i), ĥ3 (i)i. This
computation has to be correct by the choice of the function ĥ.
Since (C ⊕ Y )0 ≤T Y 0 , it follows that C 0 ≤T Y 0 .
¤
8
C. T. CHONG
We now apply Theorem 3.1 to study Ramsey’s Theorem for pairs.
The proof of the following result follows essentially Jockusch [9]. The
only point to note is that the construction works under the assumption
of BΣ02 .
Proposition 3.1. Let M be a BΣ02 model of RCA0 . Let Y ∈ X.
Then there is a Y -recursive partition fJ of [M ]2 into two colors with
no homogeneous solution recursive in Y 0 .
Definition 3.1. Let ϕ(X, Y ) be a second order formula with X and
Y as free set variables. A model M of RCA0 is a double-jump model
of ∀Y ∃Xϕ(X, Y ) if for all Y ∈ M, there is an X ∈ M such that
X ≤T Y 00 and M |= ϕ(X, Y ).
Results of Stephan and Jockusch [?] and Cholak, Jockusch and Slaman [1] show that every ω-model of RCA0 is an ω-submodel of one
that is a double-jump model of RT22 , if ∀Y ∃Xϕ(X, Y ) is interpreted
as asserting the existence of a homogeneous set X for a Y -recursive
2-coloring of pairs. The same conclusion also holds for COH. It turns
out that the strength of first order induction plays a crucial role for
this to be true.
Corollary 3.1. No BΣ02 model of RCA0 is a double-jump model of
RT22 .
Proof. Suppose otherwise. Let M be a BΣ02 model of RCA0 that is also
a double-jump model for RT22 . Let Y ∈ M be chosen such that the
Σ02 (M) cut I is defined with parameter Y . Let fJ be the Y -recursive
partition given in Proposition 3.1. Then by assumption there is a
homogeneous solution HfJ ≤T Y 00 for fJ . By Theorem 3.1, HJ <T Y 0 .
However, this contradicts Proposition 3.1.
¤
Let DB-RT22 be the following statement: For any Y , any Y -recursive
2-coloring of pairs has a homogeneous set X such that X ≤T Y 00 .
Corollary 3.2. Let M be a model of RCA0 . The following are equivalent:
(i) M |= IΣ02 ;
(ii) M is an M -submodel of an M∗ |= DB-RT22 .
Proof. The proof of (i) ⇒ (ii) may be adapted from Theorem 4.2 of
Jockusch [9], by carrying out the construction using Σ02 induction. For
the other direction, first note that Hirst [8] showed that every model
of RCA0 + RT22 satisfies BΣ02 . Suppose that M is an M -submodel
NONSTANDARD METHODS IN RAMSEY’S THEOREM FOR PAIRS
9
of M∗ |= DB-RT22 and M∗ is a BΣ02 model. Again let Y be such
that there is a Σ02 (Y ) cut with a Σ02 cofinal function from the cut into
M . Let fJ be the Jockusch partition of pairs recursive in Y . If HJ
is homogeneous for fJ and recursive in Y 00 , then by Theorem 3.1 it is
recursive in Y 0 , which is not possible. Thus M∗ , hence M, satisfies
IΣ02 .
¤
The proof of the next result is essentially in Theorem 12.4 of [1].
Proposition 3.2. Let M be a BΣ02 model of RCA0 . Let g : I → M be
Σ02 , increasing and cofinal with parameter Y . Then there is a Z ≤T Y
with a Z-array that has no Z-cohesive set C satisfying C 0 ≤T Y 0 .
0
Proof. Let e be such that ΦYe is a {0, 1}-valued partial function that
has no Y 0 -computable extension to a total function. Define Z =
Y0
{(s, t)|Φt t (s) ↓= 0}. Let f (s, t) = 0 if (s, t) ∈ Z, and equal to 1
otherwise. Let Zs = {t|(s, t) ∈ Z}. Then both f and Z are recursive
in Y .
Suppose that C is cohesive for the array hZs is∈M and C 0 ≤T Y 0 .
Then fˆ(s) = limt∈C f (s, t) exists for each s and is computable in C 0 ,
hence in Y 0 by assumption. Furthermore fˆ is a total function extending
0
ΦYe , contradicting the choice of e.
¤
Let ∀Y ∃Xϕ(X, Y ) be COH: For every Y ∈ M and Z ≤T Y , every Zarray has a Z-cohesive set X ∈ M. Theorem 3.1 leads to the following
corollary.
Corollary 3.3. Let M be a BΣ02 model of RCA0 . Then M is not a
double-jump model of COH.
It follows from Corollary 3.3 that starting with a BΣ02 model M
of RCA0 , it is not possible to add second order objects to M within
double jump to obtain a model of RCA0 + COH while preserving BΣ02 .
This is expressed as follows. Let DB-COH denote the statement that
every Y -array has a Y -cohesive set recursive in Y 00 .
Corollary 3.4. RCA0 + COH does not prove DB-COH.
Chong, Slaman and Yang [3] have recently proved a Π11 -conservation
theorem for RCA0 + COH + BΣ02 over RCA0 + BΣ02 . A key step in
the proof is to expand, for a given M and Y -array where Y ∈ M,
a Y -cohesive set in the generic extension that preserves BΣ02 . The
Y -cohesive set thus obtained is not recursive in Y 00 (in fact highly
non-effective). Corollary 3.3 hints at the need for a non-Y 00 -effective
approach.
10
C. T. CHONG
Let ∀Y ∃Xϕ(X, Y ) be SRT22 . Theorem 3.1, when applied to SRT22 ,
says that any BΣ02 double-jump model M of SRT22 is necessarily one
that contains only incomplete homogeneous sets (incomplete relative
to the parameter defining the 2-coloring). Furthermore, M satisfies
the following useful result whose origin (in the form of α-recursion
theory) goes back to Shore [13] (See also Mytilinaios and Slaman [11]
and Chong and Yang [4]):
Proposition 3.3. Let M be a BΣ02 model of RCA0 . Let I be a Σ92 (M)
cut with parameter Y ∈ M. Then every X <T Y 0 in M is Y -low,
i.e. X 0 ≤T Y 0 .
Proof. Let M and Y be as given, and let I be a Σ02 (Y ) cut. Let
X <T Y 0 be in M. Then BΣ02 (X) holds in M and so X 0 is regular.
Since X 0 ≤T Y 00 , Lemma 3.1 implies that X 0 is recursive in I ⊕ Y 0 . The
argument in the proof of Theorem 3.1 (after Claim 3) now implies that
X 0 ≤T Y 0 .
¤
It is not difficult to see that every stable 2-coloring of pairs has a
homogeneous solution ∆02 in the parameter defining the coloring. In a
BΣ02 setting, the crucial question is whether every stable 2-coloring of
pairs has a homogeneous solution that is low relative to the parameter
that defines the coloring. If the answer is yes, then one may generate
a BΣ02 model of RCA0 + SRT22 as follows: Begin with a countable BΣ02
model M0 (with no second order objects). Let this be the ground
model. At stage n + 1, construct an incomplete homogeneous set for
a stable 2-coloring defined over M
S n . One can arrange the stable 2colorings in such a way that M = n Mn is a model of SRT22 . Now BΣ02
is preserved in every Mn since only low sets are added, guaranteeing
that M |= BΣ02 . Then Proposition 3.1 ensures that M is not a model
of RT22 .
We end this paper with two questions.
Question 1. Is there a double-jump BΣ02 model for SRT22 ?
Question 2. Is there a BΣ02 model of RCA0 + RT22 or COH obtained
from a ground model by working within the nth jump, where n > 2?
NONSTANDARD METHODS IN RAMSEY’S THEOREM FOR PAIRS
11
References
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Department of Mathematics, National University of Singapore, Singapore 117542
E-mail address: [email protected]